108 research outputs found
Odd-Cycle-Free Facet Complexes and the K\"onig property
We use the definition of a simplicial cycle to define an odd-cycle-free facet
complex (hypergraph). These are facet complexes that do not contain any cycles
of odd length. We show that besides one class of such facet complexes, all of
them satisfy the
K\"onig property. This new family of complexes includes the family of
balanced hypergraphs, which are known to satisfy the K\"onig property. These
facet complexes are, however, not Mengerian; we give an example to demonstrate
this fact.Comment: 12 pages, 11 figure
Efficiently Computing Minimal Sets of Critical Pairs
In the computation of a Gr"obner basis using Buchberger's algorithm, a key
issue for improving the efficiency is to produce techniques for avoiding as
many unnecessary critical pairs as possible. A good solution would be to avoid
_all_ non-minimal critical pairs, and hence to process only a_minimal_ set of
generators of the module generated by the critical syzygies. In this paper we
show how to obtain that desired solution in the homogeneous case while
retaining the same efficiency as with the classical implementation. As a
consequence, we get a new Optimized Buchberger Algorithm.Comment: LaTeX using elsart.cls, 27 page
ON THE HILBERT QUASI - POLYNOMIALS FOR NON - STANDARD GRADED RINGS
Abstract
The Hilbert function, its generating function and the Hilbert polynomial of a graded ring K[x1, . . . , xk ] have been extensively studied since the famous paper of Hilbert: Ueber die Theorie der algebraischen Formen ([Hilbert, 1890]). In particular the coefficients and the degree of the Hilbert polynomial play an important role in Algebraic Geometry.
If the ring graduation is non-standard, then its Hilbert function is not definitely equal to a polynomial but to a quasi-polynomial.
It turns out that a Hilbert quasi-polynomial P of degree n splits into a polynomial S of degree n and a lower degree quasi-polynomial T . We have completely determined the degree of T and the first few coefficients of P . Moreover, the quasi-polynomial T has a periodic structure that we have described. We have also developed a software to compute effectively the Hilbert quasi-polynomial for any ring K[x1, . . . , xk ]/I
On the Hilbert quasi-polynomials for non-standard graded rings
The Hilbert function, its generating function and the Hilbert polynomial of a graded ring K[x1, . . . , xk ] have been extensively studied since the famous paper of Hilbert: Ueber die Theorie der algebraischen Formen ([Hilbert, 1890]). In particular, the coefficients and the degree of the Hilbert polynomial play an important role in Algebraic Geometry. If the ring grading is non-standard, then its Hilbert function is not eventually equal to a polynomial but to a quasi-polynomial. It turns out that a Hilbert quasi- polynomial P of degree n splits into a polynomial S of degree n and a lower degree quasi-polynomial T . We have completely determined the degree of T and the first few coefficients of P . Moreover, the quasi-polynomial T has a periodic structure that we have described. We have also developed a software to compute effectively the Hilbert quasi-polynomial for any ring K[x1 , . . . , xk]/I
Reducing the size and number of linear programs in a dynamic Gr\"obner basis algorithm
The dynamic algorithm to compute a Gr\"obner basis is nearly twenty years
old, yet it seems to have arrived stillborn; aside from two initial
publications, there have been no published followups. One reason for this may
be that, at first glance, the added overhead seems to outweigh the benefit; the
algorithm must solve many linear programs with many linear constraints. This
paper describes two methods of reducing the cost substantially, answering the
problem effectively.Comment: 11 figures, of which half are algorithms; submitted to journal for
refereeing, December 201
A good leaf order on simplicial trees
Using the existence of a good leaf in every simplicial tree, we order the
facets of a simplicial tree in order to find combinatorial information about
the Betti numbers of its facet ideal. Applications include an Eliahou-Kervaire
splitting of the ideal, as well as a refinement of a recursive formula of H\`a
and Van Tuyl for computing the graded Betti numbers of simplicial trees.Comment: 17 pages, to appear; Connections Between Algebra and Geometry,
Birkhauser volume (2013
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